An improved estimate for the number of zeros of Abelian integrals for cubic Hamiltonians

摘要

Suppose that the real generic cubic Hamiltonian H(x, y), (x, y) ε ?~2, possesses three saddle points and one centre. Let ∑ ? ? be the set of values h of H(x, y), for which there exists a closed component δ(h) of the level curve {H(x, y) = h}, free of critical points. In this paper, we obtain a better upper bound than previously known for the number of zeros of the Abelian integrals I (h) = ∫_(δ(h))[g(x, y) dx - f (x, y) dy] for h ε ∑ in terms of the maximum of the degrees of the polynomials f (x, y) and g(x, y).